Question

Suppose a bakery has 16 employees to be designatedas bread bakers (B) and cake bakers (C), sothat B + C = 16. Draw the production possibilitiesfrontier for bread (y) and cakes (x) for the followingproduction functions:

a. y = 2B^0.5 and x = C^0.5

b. y = B and x = 2C^0.5

Short answer

The production possibilities frontier for bread and cake is as below:

a) Production function; y = 2B^0.5 and, x = C^0.5

b) Production function; y=B and, x=2C^0.5

Step-by-step solution

Step 1. Explanation for part (a)

It is known that B+C=16. Therefore, it can also be written as B=16-C. Now, write y as:

$y=2{\left(16-C\right)}^{0.5}$

It is known:

$$\begin{gathered} x=C^{0.5} \\ C=x^2. \end{gathered}$$

Substitute for C in y to get the equation for the production possibilities frontier:

$$\begin{gathered} y=2{\left(16-x^2\right)}^{0.5} \\ or \\ y=2\sqrt{16-x^2} \end{gathered}$$

This is the equation of PPF. Plot the given equation on the graph; the production possibility frontier will be given as follows:

The graph above shows values of y on the y-axis. These are selected, keeping the highest value in mind that will be 8 as given by the equation, and the values of x have depicted upon the x-axis, where 4 is the highest.

Step 2. Increased supply of gold

It is known B=16-C; write for y as:

$y=16-B$

Now for x production function:

$C={\left(\frac{x}{2}\right)}^2$

Substituting this in the expression y:

$y=16-{\left(\frac{x}{2}\right)}^2$

This is the equation for the production possibility frontier; based on this, the graph is as below:

The y-axis represents y with 16 being the highest, and the x-axis depicts x with 8 being the highest; the production possibilities frontier is formed using the equation. Therefore, the other values are selected based on these values.